My son holds a UK passport. He is married to an american girl and has a daughter and has lived in America for around 15 years. He is a legal resident. What paperwork does he need to travel to the UK for 10 days and then back to the USA ?
To travel to the UK, your son will need his UK passport. For travel from the UK back to the US, he will need his UK passport and his US green card. No other travel paperwork should be necessary.
The only exception might be if his daughter is under 18, and he were to travel with his daughter but without his wife. Additional paperwork might be needed to prove his wife gave permission for him to travel with his daughter. This would not be necessary if he travels alone or with his whole family.
I myself am a UK citizen with a US residency and I've been travelling to the UK and back with a UK passport and my green card for 5 years. I want to advise in case your son is not vaccinated to get a test, since after it was announced that the UK no longer requires tests/vax status, they asked me for those. I also advise in case your son wants to get a citizenship in the future to not leave for more than a year as this may be a reason to reject the application.